Problem: There are $32$ forwards and $80$ guards in Leo's basketball league. Leo must include all players on a team and wants each team to have the same number of forwards and the same number of guards. If Leo creates the greatest number of teams possible, how many guards will be on each team?
Answer: In order to know how many teams Leo can create, we need a number that is a factor of ${32}$ and ${80}$, so that the ${32}$ forwards and the ${80}$ guards can be divided up evenly. So, if there were $\gray{4}$ teams, there would be ${32} \div \gray{4} = 8$ forwards and ${80} \div \gray{4} = 20$ guards on each team. This creates equal teams, but it isn't the greatest number of teams possible! To find the greatest number of teams, we want to find the greatest common factor of ${32}$ and ${80}$. To do so, let's find factors of ${32}$ and ${80}$. ${32}$ : $1, 2, 4, 8,{16}, 32$ ${80}$ : $1, 2,4, 5, 8, 10, {16}, 20, 40, 80$ The greatest common factor of ${32}$ and ${80}$ is ${16}$. In math notation this looks like: $ \text{gcf}({32},{80}) = {16}$. The greatest number of teams that Leo can make is ${16}$. To find the number of guards on each team, we need to divide the total number of guards by the number of teams: $ {80} \div {16} = 5.$ There will be $ 5$ guards on each team.